A Néron–Ogg–Shafarevich criterion for K3 surfaces. Issue 2 (27th February 2019) - Chiarellotto, Bruno; Lazda, Christopher; Liedtke, Christian
Abstract: The naive analogue of the Néron–Ogg–Shafarevich criterion is false for K3 surfaces, that is, there exist K3 surfaces over Henselian, discretely valued fields K, with unramified ℓ ‐adic étale cohomology groups, but which do not admit good reduction over K . Assuming potential semi‐stable reduction, we show how to correct this by proving that a K3 surface has good reduction if and only if H e ́ t 2 ( X K ¯, Q ℓ ) is unramified, and the associated Galois representation over the residue field coincides with the second cohomology of a certain 'canonical reduction' of X . We also prove the corresponding results for p ‐adic étale cohomology.